Skip to main content

A scaling equation of state near the critical point and the stability boundary of a liquid


A new scaling equation of state is proposed to describe the equilibrium thermodynamic properties of liquids near the critical point. In distinction from the existing scaling equations, which are parametric, the new equation is nonparametric and is expressed directly in terms of the physical quantities (pressure, temperature, and so on). It creates a number of advantages for the traditional representation and data processing. The equation gives rise to a binodal, spinodal, and a curve of thermal capacity divergence (pseudospinodal). The equation is expressed in terms of reduced variables (the ratio of the deviation of a thermodynamic variable from its critical value to the critical value) and contains 3 system-dependent adjustable constants. With the help of this equation, we conducted an approximation of the experimental PVT data in the critical region of 4He, C2H4, and H2O with a pressure error of 0.4% and carried out a calculation of the C v 4He thermal capacity with no more than 4% error using a three-system constant determined from the PVT data.

This is a preview of subscription content, access via your institution.


  1. 1.

    Patashinskii, A. Z. and Pokrovskii, V. L., Fluktuatsionnaya teoriya fazovykh perekhodov (Fluctuation Theory of Phase Transitions), Moscow: Nauka, 1982.

    Google Scholar 

  2. 2.

    Schofield, P., Parametric Representation of the Equation of State Near a Critical Point, Phys. Rev. Lett., 1969, vol. 22, no. 12, pp. 606–608.

    Article  ADS  Google Scholar 

  3. 3.

    Schofield, P., Litster, G.D., and Ho, G.T., Correlation between Critical Coefficients and Critical Exponents, Phys. Rev. Lett., 1969, vol. 23. no. 19, pp. 1098–1102.

    Article  ADS  Google Scholar 

  4. 4.

    Sartakov, A. G. and Martynets, V. G., Equation of State for a Liquid in a Wide Neighborhood of the Critical Point, Izv. Sib. Otd. AN SSSR. Ser. Khim. Nauk, 1982, issue 3, pp. 14–19.

  5. 5.

    Chen, Z.Y., Albright, P.C., and Sengers, J.V., Crossover from Singular Critical to Regular Classical Thermodynamic Behavior of Fluid, Phys. Rev. A, 1990, vol. 41, no. 6, pp. 3161–3177.

    Article  ADS  Google Scholar 

  6. 6.

    Agayan, V.A., Anisimov, M.A., and Sengers, J.V., Crossover Parametric Equation of State for Ising-like Systems, Phys.Rev. E, 2001, vol. 64, 026125-1–026125-19.

    Article  ADS  Google Scholar 

  7. 7.

    Rykov, V., Ustjuzhanin, E., Magee, J., et al, A Combined Equation of State of HFC 134a., in Proc. of Fifteenth Symposium on Thermophysical Properties, 2003, Boulder, Colorado, USA.

  8. 8.

    Rykov, V.A., Equation of State in the Critical Region Constructed within a Framework of the Method of Several “Pseudospinodal” Curves, Zh. Fiz. Khim., 1985, vol. 59, no. 10, pp. 2605–2607.

    Google Scholar 

  9. 9.

    Kaplun, A.B. and Meshalkin, A.B., On the Value of Thermal Capacity C v at the Vapor-Liquid Critical Point and in Two-Phased Region of Thermodynamic System, Dokl. Akad. Nauk, 2005, vol. 404, no. 3, pp. 329–332.

    Google Scholar 

  10. 10.

    Abdulagatov, I.M. and Alibekov, B.G., Characteristic Properties of the Isochoric Thermal Capacity Behavior for Pure Substances near the Liquid-Vapor Critical Point and a “Pseudospinodal” Hypothesis, in GSSSD. Teplofizicheskie svoistva veschestv i materialov (Thermophysical Properties of Substances and Materials), Moscow: izd. Standartov, 1985, issue 22. pp. 97–101.

    Google Scholar 

  11. 11.

    Sorensen, C.M. and Semon, M.D., Scaling Equation of State Derived from the Pseudospinodal, Phys. Rev. A, 1980, vol. 21, pp. 340–355.

    Article  ADS  Google Scholar 

  12. 12.

    Griffiths, R.B., Thermodynamic Functions for Fluids and Ferromagnets near the Critical Point, Phys. Rev., 1967, vol. 158, pp. 176–189.

    Article  ADS  Google Scholar 

  13. 13.

    Bezverhii, P.P., Martynets, V.G., and Matizen, E.V., Non-Parametric Scaling Equation of State and Approximation of P-ρ-T Data near the Vaporization Critical Point of Liquids, Zh. Eksp. Teor. Fiz., 2004, vol. 126,issue 5, no. 11, pp. 1146–1152.

    Google Scholar 

  14. 14.

    Kukarin, V.F., Martynets, V.G., Matizen, E.V., and Sartakov, A.G., Experimental Studying of P-ρ-T Dependences for He4 near the Vaporization Critical Point, Fiz. Niz. Temp., 1980, vol. 6, no. 5, pp. 549–559.

    Google Scholar 

  15. 15.

    Rivkin, S.L. and Ahundov, T.S., Experimental Studying of the Water Specific Volume at Temperatures 374–500°C and Pressures up to 600 kg/sm2, Teploenergetika, 1963, no. 10, pp. 66–71. 1

  16. 16.

    Rivkin, S.L., Ahundov, T.S., Kremenevskaya, E.A., and Asadullaeva, N.N., On the Research of the Water Specific Volume in the Region near the Critical Point, Teploenergetika, 1966, no. 4, pp. 59–71.

  17. 17.

    Rivkin, S.L. and Ahundov, T.S., Experimental Studying of the Water Specific Volume, Teploenergetika, 1962, no. 1, pp. 57–65.

  18. 18.

    Hastings, J.R., Levelt Sengers, J.M.H., and Balfour, F.W., The Critical-Region Equation of State of Ethene and the Effect of Small Impurities, J. Chem. Thermodynamics, 1980, vol. 12, no. 11, pp. 1009–1045.

    Article  Google Scholar 

  19. 19.

    Moldover, M.R., Scaling of the Specific-Heat Singularity of the 4He near Its Critical Point, Phys. Rev., 1969, vol. 182, no. 1, pp. 342–352.

    Article  ADS  Google Scholar 

  20. 20.

    Kukarin, V.F., Martynets, V.G., Matizen, E.V., and Sartakov, A.G., Approximation of P-ρ-T Data near the Critical Point of He4 by a New Equation of State, Fiz. Niz. Temp., 1981, vol. 7, no. 12, pp. 1501–1508.

    Google Scholar 

  21. 21.

    Martynets, V.G., Matizen, E.V., and Sartakov, A.G., Calorific Equation of State of a Liquid in a Wide Neighborhood of the Critical Point, Fiz. Niz. Temp., 1984, vol. 10, no. 5, pp. 503–509.

    Google Scholar 

  22. 22.

    Roach, P.R., Pressure-Density-Temperature Surface of 4He near Its Critical Point, Phys. Rev., 1968, vol. 170, no. 1, pp. 213–223.

    Article  ADS  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to E. V. Matizen.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bezverkhii, P.P., Martynets, V.G. & Matizen, E.V. A scaling equation of state near the critical point and the stability boundary of a liquid. J. Engin. Thermophys. 16, 164–168 (2007).

Download citation


  • equation of state
  • critical point
  • isochoric thermal capacity